Spectral Asymptotics for the Semiclassical Dirichlet to Neumann Operator

Andrew Hassell () and Victor Ivrii ()
Additional contact information
Andrew Hassell: Mathematical Sciences Institute, Australian National University
Victor Ivrii: University of Toronto, Department of Mathematics

Chapter Chapter 29 in Microlocal Analysis, Sharp Spectral Asymptotics and Applications V, 2019, pp 468-494 from Springer

Abstract: Abstract Let M be a compact Riemannian manifold with smooth boundary, and let $$R(\lambda )$$ be the Dirichlet-to-Neumann operator at frequency $$\lambda $$ . The semiclassical Dirichlet-to-Neumann operator $$R_{\text{scl}}(\lambda) $$ is defined to be $$\lambda^{-1} R(\lambda) $$ . We obtain a leading asymptotic for the spectral counting function for $$R_{\text{scl}}(\lambda) $$ in an interval $$[a_1, a_2) $$ as $$\lambda \to \infty $$ , under the assumption that the measure of periodic billiards on $$T^*M $$ is zero. The asymptotic takes the form $$\mathsf{N}(\lambda; a_1,a_2) = \bigl( \kappa(a_2)-\kappa(a_1)\bigr)\text{vol}'(\partial M) \lambda^{d-1}+o(\lambda^{d-1}), $$ where $$\kappa(a) $$ is given explicitly by $$\begin{aligned}\kappa(a) &= \frac{\omega_{d-1}}{(2\pi)^{d-1}} \bigg( -\frac{1}{2\pi} \int_{-1}^1 (1 - \eta^2)^{(d-1)/2} \frac{a}{a^2 + \eta^2} \, d\eta \\& \qquad\qquad\qquad\qquad - \frac{1}{4} + H(a) (1+a^2)^{(d-1)/2} \bigg) .\end{aligned} $$

Keywords: Dirichlet-to-Neumann operator; semiclassical Dirichlet-to-Neumann operator; spectral asymptotics; 35P20; 58J50 (search for similar items in EconPapers)
Date: 2019
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-30561-1_29

Ordering information: This item can be ordered from
http://www.springer.com/9783030305611

DOI: 10.1007/978-3-030-30561-1_29

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().